Geodesic
Filial rings on direct sums and direct products of torsion-free abelian groups
Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 200-212

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A ring whose additive group coincides with an abelian group $G$ is called a ring on $G$. An abelian group $G$ is called a $TI$-group if every associative ring on $G$ is filial. If every (associative) ring on an abelian group $G$ is an $SI$-ring (a hamiltonian ring), then $G$ is called an $SI$-group (an $SI_H$-group). In this article, $TI$-groups, $SI_H$-groups and $SI$-groups are described in the following classes of abelian groups: almost completely decomposable groups, separable torsion-free groups and non-measurable vector groups. Moreover, a complete description of non-reduced $TI$-groups, $SI_H$-groups and $SI$-groups is given. This allows us to only consider reduced groups when studying $TI$-groups.
Keywords: abelian group, ring on a group, filial ring, $TI$-group. 
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     author = {E. I. Kompantseva and T. K. T. Nguyen and V. A. Gazaryan},
     title = {Filial rings on direct sums and direct products of torsion-free abelian groups},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {200--212},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a14/}
}
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E. I. Kompantseva; T. K. T. Nguyen; V. A. Gazaryan. Filial rings on direct sums and direct products of torsion-free abelian groups. Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 200-212. http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a14/